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The Conrad Program: From `-Groups to Algebras of

Logic

Michal Botur

Department of Algebra and Geometry, Palacký University in Olomouc. Czech Republic

Jan Kühr

Department of Algebra and Geometry, Palacký University in Olomouc, Czech Republic

Lianzhen Liu

School of Science, Jiangnan University, China

Constantine Tsinakis∗

Department of Mathematics, Vanderbilt University, U.S.A.

Abstract

A number of research articles have established the significant role of lattice- ordered groups (`-groups) in logic. The fact that underpins these studies is the realization that important algebras of logic may be viewed as `-groups with a modal operator. These connections are just the tip of the iceberg. The purpose of the present article is to lay the groundwork for, and provide sig- nificant initial contributions to, the development of a Conrad type approach to the study of algebras of logic. The term Conrad Program refers to Paul Conrad’s approach to the study of `-groups, which analyzes the structure of individual or classes of `-groups by primarily using strictly lattice theoretic properties of their lattices of convex `-subgroups. The present article demon- strates that large parts of the Conrad Program can be profitably extended in the setting of e-cyclic residuated lattices – that is residuated lattices that satisfy the identity x\e ≈ e/x. An indirect benefit of this work is the intro-

∗Corresponding author Email addresses: michal.botur@upol.cz (Michal Botur), jan.kuhr@upol.cz (Jan

Kühr), lianzhen2003@yahoo.com (Lianzhen Liu), constantine.tsinakis@vanderbilt.edu (Constantine Tsinakis)

Preprint submitted to Journal of Algebra July 26, 2015

duction of new tools and techniques in the study of algebras of logic, and the enhanced role of the lattice of convex subalgebras of a residuated lattice.

Keywords: Residuated lattices, Hamiltonian residuated lattices, Lattice-ordered groups, GMV-algebras 2000 MSC: Primary: 06F05, Secondary: 06D35, 06F15, 03G10,03B47, 08B15

1. Introduction

There have been a number of studies providing compelling evidence of the importance of lattice-ordered groups (`-groups) in the study of algebras of logic1. For example, a fundamental result [33] in the theory of MV-algebras is the categorical equivalence between the category of MV-algebras and the category of unital Abelian `-groups. Likewise, the non-commutative general- ization of this result in [13] establishes a categorical equivalence between the category of pseudo-MV-algebras and the category of unital `-groups. Fur- ther, the generalization of these two results in [29] shows that one can view GMV-algebras as `-groups with a suitable modal operator. Likewise, the work in [29] offers a new paradigm for the study of various classes of can- cellative residuated lattices by viewing these structures as `-groups with a suitable modal operator (a conucleus).

The preceding connections are just the tip of the iceberg. Here we lay the groundwork for, and provide some significant initial contributions to, developing a Conrad type approach to the study of algebras of logic. The term Conrad Program traditionally refers to Paul Conrad’s approach to the study of `-groups, which analyzes the structure of individual `-groups, or classes of `-groups, by primarily using strictly lattice theoretic properties of their lattices of convex `-subgroups. Conrad’s papers [6–9] in the 1960s pioneered this approach and amply demonstrated its usefulness. A survey of the most important consequences of this approach to `-groups can be found

1We use the term algebra of logic to refer to residuated lattices – algebraic counterparts of propositional substructural logics – and their reducts. Substructural logics are non- classical logics that are weaker than classical logic, in the sense that they may lack one or more of the structural rules of contraction, weakening and exchange in their Genzen-style axiomatization. These logics encompass a large number of non-classical logics related to computer science (linear logic), linguistics (Lambek Calculus), philosophy (relevant logics), and many-valued reasoning.

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in [1], while complete proofs for most of the surveyed results can be found in Conrad’s “Blue Notes” [10], as well as in [2] and [11].

The present article and its forthcoming successors will demonstrate that large parts of the Conrad Program can be profitably extended in the set- ting of e-cyclic residuated lattices – that is residuated lattices that satisfy the identity x\e ≈ e/x. This variety encompasses most varieties of no- table significance in algebraic logic, including `-groups and, more generally, all cancellative varieties of residuated lattices, MV-algebras, pseudo-MV- algebras, GMV-algebras, semilinear GBL-algebras, BL-algebras, Heyting al- gebras, commutative residuated lattices, and integral residuated lattices. A byproduct of this work is the addition of new tools and techniques for study- ing algebras of logic.

Let us now be more specific about the structure and results of this pa- per. In Section 2, we recall some necessary background from the theory of residuated lattices. In Section 3, we study in detail the lattice C(L) of con- vex subalgebras of an e-cyclic residuated lattice L. The main result of this section is Theorem 3.8 which asserts that C(L) is an algebraic distributive lattice whose compact elements – the principal convex subalgebras – form a sublattice. Further, Lemma 3.2 of the same section provides an element- wise description of the convex subalgebra generated by an arbitrary subset of L. The development of the material of this section is a natural extension of the techniques used to study the lattice of normal convex subalgebras, as developed in [5].

In Section 4, we consider the role of prime convex subalgebras (meet- irreducible elements) and polars (pseudocomplements) in C(L). For example, it is shown (Lemma 4.2) that if the e-cyclic residuated lattice L satisfies the left or right prelinearity law, then a convex subalgebra H of L is prime iff the set of all convex subalgebras exceeding H is a chain under set-inclusion. Hence, see also Proposition 4.5, the lattice K(C(L)) of principal convex sub- algebras of L is a relatively normal lattice. Further, a description of minimal prime convex subalgebras in terms of polars is provided by Proposition 4.10.

Section 5 is concerned with semilinearity, and more precisely with the question of whether this property can be “captured” in the lattice of convex subalgebras. We prove in Theorem 5.6 that a variety V of e-cyclic residuated lattices that satisfy either of the prelinearity laws is semilinear iff for every L ∈ V , all (principal) polars in L are normal. Equivalently, all minimal prime convex subalgebras of L are normal. Further, Theorem 5.9 presents a characterization of the variety SemRL of semilinear residuated lattices that

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does not involve multiplication. In Section 6, we study Hamiltonian residuated lattices, that is, residuated

lattices in which convex subalgebras are normal. Theorem 6.2 characterizes Hamiltonian varieties of e-cyclic residuated lattices; more precisely, a class closed with respect to direct products is Hamiltonian iff it satisfies certain identities. While it is known that there exists a largest Hamiltonian variety of `-groups, viz., the variety of weakly Abelian `-groups, Theorem 6.3 estab- lishes that this is not the case for Hamiltonian varieties of e-cyclic residuated lattices.

In Section 7 we ask whether the lattice of convex subalgebras of a resid- uated lattice that satisfies either prelinearity law is isomorphic to the lattice of convex `-subgroups of an `-group. The main result of the section pro- vides an affirmative answer to the question when the residuated lattice is a GMV-algebra.

Section 8 offers a few suggestions for future developments in the subject.

2. Basic notions

In this section we briefly recall basic facts about the varieties of residuated lattices, referring to [5], [19], [14], and [28] for further details. These varieties provide algebraic semantics for substructural logics, and encompass other important classes of algebras such as `-groups.

A residuated lattice is an algebra L = (L, ·, \, /,∨,∧, e) satisfying:

(a) (L, ·, e) is a monoid;

(b) (L,∨,∧) is a lattice with order ≤; and

(c) \ and / are binary operations satisfying the residuation property:

x · y ≤ z iff y ≤ x\z iff x ≤ z/y.

We refer to the operations \ and / as the left residual and right residual of ·, respectively. As usual, we write xy for x · y and adopt the convention that, in the absence of parenthesis, · is performed first, followed by \ and /, and finally by ∨ and ∧.

Throughout this paper, the class of residuated lattices will be denoted by RL. It is easy to see that the equivalences that define residuation can be

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expressed in terms of finitely many equations and thus RL is a finitely based variety (see [5], [3]).

The existence of residuals has the following basic consequences, which will be used in the remainder of the paper without explicit reference.

Lemma 2.1. Let L be a residuated lattice.

(1) The multiplication preserves all existing joins in each argument; i.e., if∨ X and

∨ Y exist for X, Y ⊆ L, then

∨ x∈X,y∈Y (xy) exists and(∨

X )(∨

Y )

= ∨

x∈X,y∈Y

(xy).

(2) The residuals preserve all existing meets in the numerator, and convert existing joins to meets in the denominator, i